Evaluate the limit of y = 4-x^2 from 1 to 2 using the limit process with trapezoidal summation

To evaluate the limit of the function \( y = 4 - x^2 \) from \( x = 1 \) to \( x = 2 \) using the limit process with trapezoidal summation, we proceed as follows:

### Step 1: Divide the Interval
Consider the interval \([1, 2]\) and divide it into \(n\) subintervals, each of width \(\Delta x\). The width of each subinterval is:
\[
\Delta x = \frac{2 - 1}{n} = \frac{1}{n}
\]
The points on the interval are \( x_0 = 1 \), \( x_1 = 1 + \Delta x \), \( x_2 = 1 + 2\Delta x \), ..., \( x_n = 2 \).

### Step 2: Trapezoidal Sum Formula
The trapezoidal sum for the function over the interval \([1, 2]\) is given by:
\[
T_n = \frac{\Delta x}{2} \left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]
\]
Where \( f(x) = 4 - x^2 \).

### Step 3: Evaluate the Function at Each Point
Let's compute \( f(x) \) at each of the points \( x_0, x_1, \ldots, x_n \):
\[
f(x_k) = 4 - x_k^2 \quad \text{for } k = 0, 1, 2, \ldots, n
\]
Substituting the values:
\[
f(x_0) = 4 - 1^2 = 3, \quad f(x_n) = 4 - 2^2 = 0
\]
\[
f(x_k) = 4 - \left(1 + k\Delta x\right)^2
\]

### Step 4: Plug into the Trapezoidal Formula
Now substitute the values into the trapezoidal sum formula:
\[
T_n = \frac{1}{2n} \left[3 + 2\sum_{k=1}^{n-1} \left(4 - \left(1 + \frac{k}{n}\right)^2\right) + 0\right]
\]

### Step 5: Take the Limit as \( n \to \infty \)
Finally, take the limit as \( n \) approaches infinity:
\[
\lim_{n \to \infty} T_n = \int_1^2 \left(4 - x^2\right) dx
\]
To find this integral:
\[
\int_1^2 \left(4 - x^2\right) dx = \left[4x - \frac{x^3}{3}\right]_1^2
\]
\[
= \left(8 - \frac{8}{3}\right) - \left(4 - \frac{1}{3}\right) = \left(\frac{24}{3} - \frac{8}{3}\right) - \left(\frac{12}{3} - \frac{1}{3}\right)
\]
\[
= \frac{16}{3} - \frac{11}{3} = \frac{5}{3}
\]

So, the limit of the area under the curve using trapezoidal summation is \( \frac{5}{3} \) or approximately \( 1.667 \).

Would you like more details, or do you have any questions?

Here are five related questions for further exploration:
1. How does the trapezoidal rule compare with Simpson's rule in terms of accuracy?
2. Can the trapezoidal rule be used for functions that are not continuous? What changes are needed?
3. What is the geometric interpretation of the trapezoidal rule?
4. How would the approximation change if you used a smaller \( n \) in the trapezoidal rule?
5. What are the errors associated with the trapezoidal rule, and how can they be minimized?

**Tip:** The accuracy of the trapezoidal rule improves as the number of subintervals \( n \) increases, and it is generally more accurate for smooth functions.

Learn how to evaluate the limit of the function y = 4 - x^2 from x = 1 to x = 2 using the trapezoidal summation method. This problem covers key concepts in integral calculus, numerical methods, and the application of the trapezoidal rule. Suitable for college-level students and enthusiasts of calculus.

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